An insulated box containing a monoatomic gas of molar mass (M) moving with a speed `v_0` is suddenly stopped. Find the increment is gas temperature as a result of stopping the box.

Explanation

To solve the problem of finding the increment in gas temperature when an insulated box containing a monoatomic gas is suddenly stopped, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- The box containing the monoatomic gas is moving with an initial speed \( v_0 \).
- The molar mass of the gas is \( M \).

2. **Understand the Energy Transformation**:
- When the box is suddenly stopped, the kinetic energy of the gas is converted into internal energy (thermal energy) of the gas.
- The decrease in kinetic energy will equal the increase in internal energy.

3. **Calculate the Initial Kinetic Energy**:
- The kinetic energy (KE) of the gas can be expressed as:
\[ KE = \frac{1}{2} m v_0^2 \]
- Here, \( m \) is the mass of the gas. Since the molar mass \( M \) is given, we can express the mass \( m \) in terms of the number of moles \( n \):
\[ m = n \cdot M \]
- Therefore, the kinetic energy can be rewritten as:
\[ KE = \frac{1}{2} (n \cdot M) v_0^2 \]

4. **Relate Kinetic Energy to Internal Energy**:
- The increase in internal energy (\( \Delta U \)) for a monoatomic gas can be expressed as:
\[ \Delta U = \frac{nF}{2} R \Delta T \]
- For a monoatomic gas, the degrees of freedom \( F = 3 \). Thus, the equation becomes:
\[ \Delta U = \frac{3}{2} n R \Delta T \]

5. **Set Kinetic Energy Equal to Internal Energy**:
- Since the decrease in kinetic energy equals the increase in internal energy, we can write:
\[ \frac{1}{2} (n \cdot M) v_0^2 = \frac{3}{2} n R \Delta T \]

6. **Simplify and Solve for \( \Delta T \)**:
- Cancel \( n \) from both sides (assuming \( n \neq 0 \)):
\[ \frac{1}{2} M v_0^2 = \frac{3}{2} R \Delta T \]
- Rearranging gives:
\[ \Delta T = \frac{M v_0^2}{3 R} \]

7. **Final Expression**:
- The increment in gas temperature as a result of stopping the box is:
\[ \Delta T = \frac{M v_0^2}{3 R} \]